TO MATHS DAY

I. History of OrigamiII. Terms related to origamiIII. Origami and Mathematics

(Some neat theorms )IV. Constructing Polygons (Yet

another neat theorem)V. Constructing Polyhedra

(Modular Origami)VI. Other ways how maths is

used in origami

CONTENTS

•In ancient Japanese ori literally translates to folded while gami literally translates to paper. Thus the term origami translates to folded paperOrigami has roots in several different cultures. The oldest records of origami or paper folding can be traced to the Chinese. The art of origami was brought to the Japanese via Buddhist monks during the 6th century.

The Spanish have also practiced origami for several centuries.Early origami was only performed during ceremonial occasions (i.e. weddings, funerals, etc.).

HISTORY OF ORIGAMI

FLAT FOLD – An origami which you could place flat on the ground and compress without adding new creases.

CREASE PATTERN – The pattern of creases found when an origami is completely unfolded.

VERTEX – A point on the interior of the paper where two or more creases intersect.

MOUNTAIN CREASE – A crease which looks like a mountain or a ridge.

VALLEY CREASE – A crease which looks like a valley or a trench.

TERMS

The difference between the number of mountain creases and the number of valley creases intersecting at a particular vertex is always…

MAEKAWA’S THEORM

• The all dashed lines represent mountain creases while the dashed/dotted lines represent valley creases.

Let M be the number of mountain creases at a vertex x.

Let V be the number of valley creases at a vertex x.

Maekawa’s Theorem states that at the vertex x,

M – V = 2or

V – M = 2

Note – It is sufficient to just focus on one vertex of an origami.

Let n be the total number of creases intersecting at a vertex x. If M is the number of mountain creases and V is the number of valley creases, then

n = M + V

Proof of Maekawa’s Theorem(Jan Siwanowicz – 1993)

1. Take your piece of paper and fold it into an origami so that the crease pattern has only one vertex.

2.Take the flat origami with the vertex pointing towards the ceiling and fold it about 1½ inches below the vertex.

3.What type of shape is formed when the “altered” origami is opened? polygon

4. As the “altered” origami is closed, what happens to the interior angles of the polygon?

5. Some get smaller – Mountain Creases

Some get larger – Valley Creases

polygon

When the “altered” origami is folded up, we have formed a FLAT POLYGON whose interior angles are either:

0° – Mountain or360° – Valley Creases

Recap – Viewing our flat origami we have an n-sided polygon which has interior angles of measure:

0° – M of these360° – V of these

Thus, the sum of all of the interior angles would be:

0M + 360V

What is the sum of the interior angles of any polygon

sShape Angle sumside

3

5

180

180(5) – 360 or 540

PROOF OF MAEKAWA’S THEOREM(JAN SIWANOWICZ – 1993)

What is the sum of the interior angles of any polygon?

SIDES SHAPE ANGLE SUM

n(180n – 360)°

or180(n – 2)°

So, we have that the sum of all of the interior angles of any polygon with n sides is:

180(n – 2)

But, we discovered that the sum of the interior angles of each of our FLAT POLYGONS is:

0M + 360V

where M is the number of mountain creases and V is the number of valley creases at a vertex x.

Equating both of these expressions we get:

180(n – 2) = 0M + 360V

Recall that n = M + V.

So, we have:180(M + V – 2) = 0M + 360V180M + 180V – 360 = 360V

180M – 180V = 360M – V = 2

Thus, we have shown that given an arbitrary vertex x with M mountain creases and V valley creases, either:

M – V = 2or

V – M = 2

This completes our proof!

A quick view

Constructing polyhedra

POLYHEDRON – A solid constructed by joining the edges of many different polygons. (Think 3-Dimensional polygon.)

0SONOBE – A flat origami which when pieced together with identical SONOBE units can be used to modularly construct polyhedra.

SONOBE

OTHER USES OF MATHS IN ORIGAMY

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So friends what ever where ever maths is with us

in the form of origami or any

other..

BY –SAMPATH PATTJOSHI